Various masaail of Jamaat Salaah and Imaamat

Q: If the Imaam in the Haram reads a sajdah aayat in maghrib, and the musallees not knowing the imaam went into sajdah, instead go into ruku, and thereafter they join the imaam again in the remainder qiraat, is the salaah of the musallees valid?

Q: Namaz behind such imam: 

If there's a Barelvi masjid in my area and they say the following before or after azan: "assalatu wassalamu alaika ya rasulullah wa ala alika wa ashabika ya noor Allah" 

Can I pray behind them and if not, should I repeat every prayer behind them?

Q: I have a situation where in my work place in Saudi Arabia (far from any musjid in construction site), the person that often is imaam of the small musallah we perform zohr and asr salah, makes masah over his normal cotton socks when he makes wudu. 

I read salah for few months behind him and only found out about this recently. 

He confirmed that some scholars in Saudi passed a fatwa that its permissible. It would seem that many of the general public here follow this fatwa. 

I do not want to argue and create fitnah based on the situation. I know the fatwa is that wudu is not valid if masah is made over normal cotton socks based on majority opinion. 

However, based on my situation/ the fact that I did not know or could not control the imaam, is my previous salah valid or should I repeat it? 

Also, going forward, what should I do: Can I read salah behind him, based on the fact that I cannot control the imaam - perhaps consider him a different madhhab/ different opinion with salah being valid Insha Allah or is salah invalid for sure in which case. Should I read salah behind him and repeat it or should I try to avoid and read on my own. This will result in me not being able to make salah in jamaat at work and possibly create division. (Also reading salah and repeating is a challenge especially asr based on limited time and people observing you).

Q: If there are a few musjids in an area, which musjid will be the best to perform salaah in? The oldest musjid or the musjid with the most musallis?

Q: Does a muqtadee have to recite the takbeers in salaah when going from one posture to another?

Q: A late comer joins me in salaah while I'm performing my sunnah salaah. Is it permissible for me to change my intention of performing sunnah salaah alone to performing in jamaat?

Q: We have a small musallah in our area. My question is, I am hoping to perform salaah as the imaam with 1, 2 to 4 musallies every day, but most of the time, I'm reading salaah alone as the imaam in front. Will I still get the full reward of salaah with jamaat?

Q: Can you run for salaah if you're late? I have seen prominent people doing this. Not outright running, but a very fast walk that a person will have to run after them if they want to catch up.

Q: There's a few brothers in a masjid who usually don't like when any scholars from different localities come to their masjid, when jamaat comes and even their own Imaams. They find fault in them. For example, they say that one person is reading too fast, another is making ruku and sajdah too long and they only come when the Imaam has to go on the mimbar for the khutbah.

Will their Salaah be accepted like how they don't appreciate their Imaams? Please advise as to what the normal brothers can do and also what the Imaams can do.